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Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays

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  • Karaoglu, Esra
  • Merdan, Huseyin

Abstract

In this paper we give a detailed Hopf bifurcation analysis of a ratio-dependent predator–prey system involving two different discrete delays. By analyzing the characteristic equation associated with the model, its linear stability is investigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. Furthermore, some of the bifurcation properties including direction, stability and period are given. Finally, theoretical results are supported by some numerical simulations.

Suggested Citation

  • Karaoglu, Esra & Merdan, Huseyin, 2014. "Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays," Chaos, Solitons & Fractals, Elsevier, vol. 68(C), pages 159-168.
    • Handle: RePEc:eee:chsofr:v:68:y:2014:i:c:p:159-168
    DOI: 10.1016/j.chaos.2014.07.011
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    1. Li, Kai & Wei, Junjie, 2009. "Stability and Hopf bifurcation analysis of a prey–predator system with two delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2606-2613.
    2. Zhou, Xiaobing & Wu, Yue & Li, Yi & Yao, Xun, 2009. "Stability and Hopf bifurcation analysis on a two-neuron network with discrete and distributed delays," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1493-1505.
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    Cited by:

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    2. De Cesare, Luigi & Sportelli, Mario, 2020. "Stability and direction of Hopf bifurcations of a cyclical growth model with two-time delays and one-delay dependent coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. De Cesare, Luigi & Sportelli, Mario, 2022. "A non-linear approach to Kalecki’s investment cycle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 57-70.
    4. Loretti I. Dobrescu & Mihaela Neamtu & Gabriela Mircea, 2016. "Asset Price Dynamics in a Chartist-Fundamentalist Model with Time Delays: A Bifurcation Analysis," Discrete Dynamics in Nature and Society, Hindawi, vol. 2016, pages 1-15, February.
    5. Bilazeroğlu, Ş. & Merdan, H., 2021. "Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    6. Serdal Pamuk & Irem Cay, 2017. "Turing Analysis of a Mathematical Model for Interaction between Tumor Cell and Its Inhibitor," Academic Journal of Applied Mathematical Sciences, Academic Research Publishing Group, vol. 3(5), pages 46-51, 05-2017.
    7. Bilazeroğlu, Ş. & Göktepe, S. & Merdan, H., 2023. "Effects of the random walk and the maturation period in a diffusive predator–prey system with two discrete delays," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    8. Sportelli, Mario & De Cesare, Luigi, 2019. "Fiscal policy delays and the classical growth cycle," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 9-31.

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